On the IaS relation of the

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On the Ia–S relation of the SCS-CN method
M.K. Jain1, S.K. Mishra2, P. Suresh Babu3 and K. Venugopal3
1Corresponding author. National Institute of Hydrology, Roorkee-247 667, U.A., India.
E-mail: jain.mkj@gmail.com
2 Indian Institute of Technology, Roorkee-247 667, U.A., India
3 Institute of Remote Sensing, Anna University, Chennai-600 025, India
Received 2 February 2005; accepted in revised form 29 March 2006
Abstract The initial abstraction (Ia) versus maximum potential retention (S) relation in the Soil Conservation
Service Curve Number (SCS-CN) methodology was revisited, and a new non-linear relation incorporating storm
rainfall (P) and S was proposed and tested on a large set of storm rainfall-runoff events derived from the water
database of United States Department of Agriculture-Agriculture Research Service (USDA-ARS). Employing
root mean square error (RMSE), the performance of both the existing and proposed models was evaluated using
the complete database, and for model calibration and validation, data were split into two groups: based on
ordered rainfall (P-based) and runoff (Q-based). A specific formulation of the proposed model Ia ¼
lSðP=ðP þ SÞÞa with l ¼ 0.3 and a ¼ 1.5 was found to generally perform better than the existing Ia ¼ 0.2S, and
therefore was recommended for field applications. When evaluated using the observed Ia data, the proposed
version performed significantly better than the existing one.
Keywords Agriculture Research Service; curve number; initial abstraction; rainfall-runoff modelling; SCS-CN
method
Introduction
The US Soil Conservation Service-Curve Number (SCS-CN) method (Soil Conservation
Service 1956, 1964, 1969, 1971, 1972, 1985, 1993) (also known as the Natural Resource
Conservation Service Curve Number (NRCS-CN) method) is one of the most popular
methods for computing the volume and peak quantiles of direct surface runoff from rainfall
quantiles (McCuen 1982; Hjelmfelt 1991; Ponce and Hawkins 1996; Mishra and Singh
2003). Since the method works with mean and it does not account for storm duration (Ponce
and Hawkins 1996), the method is poorly suited for use in estimating runoff volumes for
actual events. However, based on a tremendous amount of experimental work, it has been
widely used in the United States and across the world and has more recently been integrated
into several rainfall-runoff models, supporting its robustness and popularity (Michel et al.
2005). Mishra and Singh (2003) provided the current state of the art of the SCS-CN
methodology, its enhanced analytical treatment, and applications to areas other than the
originally intended use.
This SCS-CN method is based on the water balance equation and two fundamental
hypotheses which can be expressed, respectively, as
P ¼ Ia þ F þ Q ð1Þ
Q
P2 Ia
¼
F
S
ð2Þ
Ia ¼ lS ð3Þ
where P ¼ total rainfall, Ia ¼ initial abstraction, F ¼ cumulative infiltration, Q ¼ direct
surface runoff, S ¼ potential maximum retention (0,1), and l ¼ initial abstraction
doi: 10.2166/nh.2006.011
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coefficient. Combination of Equations (1) and (2) leads to the popular form of the SCS-CN
method:
Q ¼
ðP2 IaÞ
2
P2 Ia þ S
ð4Þ
Equation (4) is valid for P $ Ia, otherwise, Q ¼ 0. Parameter S in Equation (4) is
expressed as
S ¼
25400
CN
2 254 ð5Þ
where S is in mm and CN is a non-dimensional quantity varying in the range (0, 100).
Since its inception, the above method has been in question for its dealing with various
aspects, such as the antecedent moisture condition (AMC), initial abstraction ratio (called
“Lambda” (l)), applicability to watersheds’ size and type, etc. Out of these, the standard
assumption of l ¼ 0.2 in Equation (3) has been frequently questioned for its validity and
applicability (Hawkins et al. 2002), invoking a critical examination of the Ia–S relationship
for pragmatic applications. Thus, the objective of this paper is (a) to provide a critical review
of the Ia–S relationship and propose an improvement incorporating P as well as S; and (b) to
compare the performance of the formulated model and its versions on a large set of observed
rainfall-runoff data, split as well as complete data sets, and the observed Ia data set.
The Ia–S relationship: an overview
An elementary expression for the transformation of rainfall into runoff isQ ¼ P–L, whereQ
is the runoff, P is the rainfall, and L is the hydrologic abstraction, which is difficult to
quantify in nature (Ponce and Hawkins 1996). This abstraction includes interception,
storage, surface storage, evaporation from water bodies such as ponds, lakes, etc.,
infiltration, and evapotranspiration from all types of vegetation. The initial abstraction (Ia)
accounts for depression (surface) storage, interception, and infiltration, occurring before
runoff begins. The interception and depression storage values vary widely with types of
vegetative cover, wind, area of depression and depth, etc., which are difficult to quantify
accurately. Hjelmfelt (1991) pointed out that many storm and landscape factors interact to
define the initial abstraction.
The initial abstraction was not a part of the SCS-CN model in its initial formulation
(Plummer and Woodward 2002), but as development continued, it was included as a fixed
ratio of Ia to S (Equation (3)). This relationship was justified on the basis of measurements in
watersheds (of less than 10 acres) despite a considerable scatter in the resulting Ia–S plot
(SCS 1985). NEH-4 (SCS 1985) reported 50% of data points to lie within 0.095 # l # 0.38,
leading to a standard value of l ¼ 0.2 (Ponce and Hawkins 1996). Because of this large
variability, the Ia ¼ 0.2S relation has been focus of discussion and modification since its
inception. For example, based on their field data, the Central Unit for Soil Conservation
(1972) recommended a l value of 0.3 for all regions of India, except for the black cotton
region for which it is 0.1 under AMC II (normal) and III (wet) conditions. Aron et al. (1977)
suggested l # 0.1 and Golding (1979) provided curve number dependent l values for urban
watersheds: l ¼ 0.075 for CN # 70, l ¼ 0.1 for 70 , CN # 80, and l ¼ 0.15 for
80 , CN # 90. Springer et al. (1980) found l ¼ 0.2 not to be appropriate for both arid and
humid watersheds and cautioned against its use for other watersheds. In his mathematical
treatment, Chen (1981) physically signified l as a square root ratio of initial ( fo) and final
infiltration rates ( f1), and varied it from 0 to 1. His Ia–S analysis was however based on ‘S
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leo_s
Comentário do texto
Uma expressão elementar para a transformação da chuva em escoamento é Q = P– L, onde Q é o escoamento, P é a chuva e L é a abstração hidrológica, que é difícil de quantificar na natureza (Ponce e Hawkins 1996). Essa abstração inclui interceptação, armazenamento, armazenamento de superfície, evaporação de corpos d'água como lagoas, lagos, etc., infiltração e evapotranspiração de todos os tipos de vegetação. A abstração inicial (Ia) é responsável pelo armazenamento, interceptação e infiltração da depressão (superfície), ocorrendo antes do início do escoamento. Os valores de armazenamento de interceptação e depressão variam amplamente com os tipos de cobertura vegetal, vento, área de depressão e profundidade, etc., que são difíceis de quantificar com precisão. Hjelmfelt (1991) apontou que muitos fatores de tempestade e paisagem interagem para definir a abstração inicial.
leo_s
Comentário do texto
A abstração inicial não fazia parte do modelo SCS-CN em sua formulação inicial (Plummer e Woodward 2002),mas conforme o desenvolvimento continuou, foi incluída como uma razão fixa de Ia para S (Equação (3)). Essa relação foi justificada com base em medições em bacias hidrográficas (de menos de 10 acres), apesar de uma dispersão considerável no gráfico Ia-S resultante (SCS 1985). NEH-4 (SCS 1985) relatou que 50% dos pontos de dados estão dentro de 0,095 <= l <= 0,38, levando a um valor padrão de l = 0,2 (Ponce e Hawkins 1996). Por causa dessa grande variabilidade, a relação Ia = 0.2S tem sido foco de discussão e modificação desde seu início. Por exemplo, com base em seus dados de campo, a Unidade Central para Conservação do Solo (1972) recomendou um valor Ia de 0,3 para todas as regiões da Índia, exceto para a região de algodão preto para a qual é 0,1 sob AMC II (normal) e III (úmido ) condições. Aron et al. (1977) sugeriu l = 0,1 e Golding (1979) forneceu valores l dependentes do número da curva para bacias hidrográficas urbanas: l = 0,075 para CN <= 70, l = 0,1 para 70 <= CN <= 80 e l = 0,15 para 80 < = CN <= 90. Springer et al. (1980) descobriram que l = 0,2 não é apropriado para bacias hidrográficas áridas e úmidas e alertaram contra seu uso em outras bacias hidrográficas. Em seu tratamento matemático, Chen (1981) significou fisicamente l como a razão da raiz quadrada das taxas de infiltração inicial (fo) e final (f1), e a variou de 0 a 1.
leo_s
Comentário do texto
Desde o seu início, o método acima tem sido questionado por lidar com vários aspectos, como a condição de umidade antecedente (AMC), razão de abstração inicial (chamada de “Lambda” (l)), aplicabilidade ao tamanho e tipo de bacias hidrográficas, etc. Destes, a suposição padrão de l = 0,2 na Equação (3) tem sido freqüentemente questionada por sua validade e aplicabilidade (Hawkins et al. 2002), invocando um exame crítico da relação Ia-S para aplicações pragmáticas. Assim, o objetivo deste artigo é (a) apresentar uma revisão crítica da relação Ia – S e propor uma melhoria incorporando tanto P quanto S; e (b) comparar o desempenho do modelo formulado e suas versões em um grande conjunto de dados de chuva-vazão observados, divididos, bem como conjuntos de dados completos e o conjunto de dados Ia observado.
including Ia’. Since S does not include Ia, the US National Engineering Handbook was
corrected accordingly (Van Mullem et al. 2002).
Cazier and Hawkins (1984) found l ¼ 0.0 to fit best to their data set and, as an alternative,
Bosznay (1989) suggested to treat Ia as a random variable. According to Ponce and Hawkins
(1996), the fixing of l at 0.2 is tantamount to regionalization based on geologic and climatic
settings. Plummer and Woodward (2002) described the origin of Ia/S and explained its
development between Ia ¼ 0 and Ia ¼ 0.2S, a result of the cooperative effort of the US Forest
Service (FS), ARS, and NRCS.
Based on their mathematical analysis, Mishra and Singh (1999a, b) found l to vary from 0
to 1. While explaining the SCS-CN’s proportionality concept (Equation (2)) using the
volumetric concept of soil–water–air, Mishra and Singh (2003) described l as the degree of
atmospheric saturation. They also provided a complete SCS-CN-based initial abstraction
model incorporating all its segments described above. For infiltration only, l ¼ fotp/S or atp,
where fo is the initial infiltration rate, tp is the time to ponding, and a is the infiltration decay
parameter analogous to the Horton (1938) infiltration decay parameter.
Among others (for example, Woodward et al. 2002; Bonta 1997), Hawkins et al. (2002)
examined the data-supported values of the Ia/S ratio and suggested accommodations for
updating its role employing two techniques, event analysis andmodel fitting, to determine Ia/S
from field data. They used only “large” storms to avoid the biasing effects of small storms
towards high CNs. They employed both “natural” and “ordered” data sets to establish a
relation between S0.05 and S0.20 and then convert CNs based on l ¼ 0.2 to those based on
l ¼ 0.05. Here, the subscript 0.05 and 0.2 of S represent l values. Sincel varied from storm to
storm or watershed to watershed, l ¼ 0.05 fitted better than l ¼ 0.2, which was found to be
unusually high. However, the values of median l varied from 0.0005 to 0.4910, with amedian
of 0.0476 for the data sets in event analysis. In model fitting, it varied from 0–0.996, with a
median of 0 for “natural” data sets, and 0–0.9793 with a median of 0.0618 for “ordered” data
sets. l ¼ 0.05 was found to significantly affect the low P/S situations, and subsequently,
hydrograph structures defined by time and peak values (especially at lower CNs).
It is interesting to note that (from the NRCS website): “. . .each relationship of Ia to S
requires a unique set of runoff curve numbers. Simple revision of the relationship of Ia to S to
something other than Ia ¼ 0.2S requires more than a simple change of the runoff equation.
There is no linear relationship between the runoff curve numbers for the two Ia conditions. It
also requires a new set of runoff curve numbers developed from analysis of small watershed
data. . ..”. Thus, it is probably not justifiable to tweak this relationship as part of the
development of a design hydrograph for a site with no calibration data available. If the
existing relationship is played with, the standard CN values are no longer valid (Rallison and
Miller 1981). Here, it is noted that Ia values were originally derived from rainfall-runoff plots
considering Ia as the rain that fell prior to the beginning of direct runoff (SCS 1985). The
sources of error associated with these estimates and listed in NEH-4 include the likelihood of
some abstracted rainfall to have eventually appeared at the outlet. For example, it is likely
that some of this rainfall might have contributed to quick response in tile-drained watersheds
(Walker et al. 1998). It was for this reason that Rallison (1980) did not recommend its further
refinement.
Proposed Ia–S relation and model formulation
It is clear from the above that the selection of a proper l value or an Ia–S relationship is
crucial to accurate estimation of direct runoff from the SCS-CN method. From Equation (3),
since Ia is related to S linearly, l take values in the range (0, 1), for S values range (0,1),
consistent with the Mishra and Singh (1999a, 2003) description. Thus, among others, it is
also the assumption of linearity in the Ia–S relation which precludes the existing SCS-CN
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methodology from being perfectly predictive of direct runoff volume. From the
mathematical treatment of Mishra and Singh (1999a) and Mishra et al. (2003), it can be
seen that l varies with C ( ¼ Q/P) and the ratio Ia/P. Since C directly reflects the variation of
CN (or S) (the larger the CN, the larger C, and vice versa), l is implicitly correlated with S
and P, rather than S alone. Furthermore, being a regional and climatic parameter, l ¼ 0.2 is
not appropriate for the watersheds other than those used in its derivation. Since precipitation
(P) in a way accounts for such climatic/meteorological characteristics of a watershed, the
proposed Ia–S relation incorporates it as follows:
Ia ¼ lS
P
Pþ S
� �a
: ð6Þ
where a ¼ a coefficient. Since Equation (6) reduces to Equation (3) for l ¼ 0.2 and a ¼ 0,
the former becomes a generalized form of the latter. Equation (6), when coupled with the
water balance equation (Equation (1)) and proportionality hypothesis (Equation (2)), leads to
a modified version of the existing SCS-CN methodology.
For testing, Table 1 shows six variants of SCS-CN-inspired models formulated using
Equations (3) and (6). In this table, CN is taken as a varying parameter, consistent with the
works of Hjelmfelt et al. (1981) and Hjelmfelt (1991) suggesting CN as a random variable.
Model M1 represents the existing SCS-CN method with Ia ¼ lS (Equation (3)) relationshipand l is allowed to vary. In Model M2, l ¼ 0.2. Models M4 and M5 are particular forms
derived from Model M3 utilizing the proposed Ia–S relationship. Models M4 and M5 with
different, but fixed, values of “l” and “a” derived by trial and error. Model M6 also
represents Equation (3) with l ¼ 0.05 (Hawkins et al. 2002).
To determine model parameters, the Marquardt (1963) algorithm of constrained least
squares approach was employed using the objective function
XN
i¼1
Q2 Q̂
� �2
i
) Minimum ð7Þ
where N is the total number of rainfall-runoff events for a catchment, i is an integer varying
from 1 to N, Q is the observed runoff from the data set, and Q̂ is the model computed runoff.
For model performance evaluation, the root mean square error (RMSE) was considered. It
is expressed as
RMSE ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
N
XN
i¼1
Q2 Q̂
� �2
i
vuut ð8Þ
where RMSE is an index of the variance between computed and observed runoff values
(Madsen et al. 2002; Itenfisu et al. 2003; Moradkhani et al. 2004).
Table 1 Various model formulations considered for the analysis
Parameters
Model a l CN Remarks
M1 0 Varying
M2 0 0.2 Existing Ia–S relation; Equation (3)
M3 Varying Varying Varying
M4 1.5 Varying Modified Ia–S relation; Equation (6)
M5 1.5 0.3
M6 0 0.05 Hawkins et al. (2002) Ia–S relation; Equation (3)
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Model calibration and validation
Data
For model testing, the data were derived from the US Department of Agriculture-
Agricultural Research Service (USDA-ARS) Water Database, a collection of rainfall and
stream flow data derived from agricultural watersheds of the United States. This database is
available on WWW at URL: http://www.ars.usda.gov/arsdb.html and http://hydrolab.
arsusda.gov/arswater.html. There are currently about 16 600 station years of data stored in
the database. The existing raingauge networks range from one station per watershed to over
200 stations. However, only 1 raingauge for each watershed was considered in this study, and
data for 22 392 storm events from 84 watersheds varying from 0.17–72 ha were used.
The available dataset for each watershed was split into two groups such that the length of
data in both calibration and validation was nearly the same and each dataset belonged to the
same population. The data were split following two schemes: based on observed
precipitation (P) values (P-based), which is an implicit descriptor of meteorological
characteristics and widely used in P–Q ordering (Hawkins et al. 2002), and based on the
observed runoff (Q) (Q-based), an actual descriptor of the runoff-producing feature of
watershed. To that end, the data were first sorted based on P (or Q) in descending order, and
then alternate rainfall-runoff values were selected and put in two groups, one for calibration
and the other for validation, for each watershed.
Parameter estimation
Employing the above Marquardt algorithm, model parameters were estimated utilizing the
dataset marked for calibration. Here, the model parameters are treated to be independent of
each other, which is a limitation of the study. For all the models studied, the parameter CN
was varied in the range (1, 100) with initial value of 80. In Model M1, the value of l was
varied in the range (0, 1) with an initial value equal to 0.05. The values of l and a in Model
M3 ranged from (0, 50) and (210, 10), respectively, with their initial value as 1.0. a ¼ 1.5 in
Models M4 and M5 and a ¼ 0.3 in Model M5 are specific cases of Model M3. Model M6
with a ¼ 0 and l ¼ 0.05 represents the Hawkins et al. (2002) model. Table 2 summarizes
the resulting parameter values in calibration. In this table, as an example, the optimized
parameter CN for Model M1 ranged from 58.67 to 91.43 with mean ¼ 81.19,
median ¼ 82.73, standard deviation ¼ 6.95, and negative skewness ¼ 1.25. Likewise, the
variation of other model parameters can be explained. The following text presents the model
calibration and validation using the above P- and Q-based datasets and compares the model
performance based on RMSE.
“P-based” dataset. The overall comparative performance of a model is judged based on the
mean RMSE value resulting from a model application to all watersheds as well as ranked
RMSE values for individual watersheds. Figure 1 depicts the scatter of RMSE values
resulting from the application of all six models to the “P-based” calibration dataset. The
resulting average RMSE values of Models M1 to M6 are 4.94, 5.18, 4.80, 4.91, 5.01, and
5.04mm, respectively. Obviously, the proposed model M3 performs the best of all other
model variants. Based on these RMSE values, the decreasing order of performance of all the
models can be given as follows:
M3 . M4 . M1 . M5 . M6 . M2:
Here, the left-hand side model works better than the adjacent right-hand side model.
For quantitative evaluation, the models were ranked (or graded) based on RMSE statistics
derived from the data of eachwatershed. To this end, ranks of 1 to 6were assigned to the lowest
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http://www.ars.usda.gov/arsdb.html
http://hydrolab.arsusda.gov/arswater.html
http://hydrolab.arsusda.gov/arswater.html
Table 2 Summary of the optimized parameters of various models in calibration of “P-based” as well as “Q-based” datasets
M1
M2
M3 M4
M5 M6
Dataset Statistics l CN CN l a CN l CN CN CN
“P-based” Min 0.00 58.67 21.72 0.00 210.00 45.90 0.13 52.58 29.20 31.81
Max 1.00 91.43 94.38 50.00 10.00 99.03 5.19 96.96 90.19 88.61
Mean 0.08 81.19 69.97 14.14 3.89 85.04 1.01 82.80 72.84 73.09
Median 0.00 82.73 74.08 3.51 2.90 88.16 0.83 84.22 77.49 75.85
STDV 0.19 6.95 15.98 19.78 3.19 11.32 0.86 9.16 14.00 10.92
Skewness 3.46 21.25 21.18 1.23 20.24 20.93 1.83 20.77 21.42 21.40
“Q-based” Min 0.00 8.47 60.20 0.00 210.00 35.79 0.00 46.58 23.61 42.20
Max 0.79 91.73 92.27 50.00 10.00 99.13 3.23 96.47 90.35 89.09
Mean 0.06 71.57 81.82 10.34 3.50 85.17 0.99 83.52 74.23 74.17
Median 0.01 74.84 82.90 3.21 2.89 91.07 0.95 86.98 77.40 75.52
STDV 0.14 14.46 6.30 16.28 3.00 13.81 0.69 10.02 12.81 9.78
Skewness 3.63 21.76 21.24 1.95 20.40 21.47 0.69 21.37 21.63 21.26
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to highest (ascending order) RMSE values, and corresponding marks of 6 to 1 were given to
each watershed. As an example, a model showing the minimum RMSE for a watershed was
ranked 1, scoring 6 marks. Figure 2 shows the ranking of models for all 84 watersheds.
Based on the above scores secured by each model in its application to the data of each
watershed, the total percent scores and corresponding overall ranking were determined, as
shown in Table 3. Apparently, the proposed model M3 performs the best based on RMSE
scoring, and the existing Model M2 the poorest. Based on overall ranking, the decreasing
order of performance of the models is as follows:
M3 . M4 . M1 . M5 . M6 . M2:
0
1
2
3
4
5
6
7
R
an
ki
ng
M1 M2 M3 M4 M5 M6
Figure 2 Models ranking based on RMSE for “P-based” dataset of calibration
0
1
2
3
4
5
6
7
8
9
10
11
12
13
R
M
SE
 (
m
m
)
M1 M2 M3 M4 M5 M6
Figure 1 Statistical comparison of models based on RMSE criteria for “P-based” dataset of calibration
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This is similar to the performance of models based on “mean” RMSE criteria. As above,
the proposed Model M3 (three-parameter model) performs the best in calibration, with high
scores (.95%) in RMSE-based evaluation of all models. It is noted that the two-parameter
Model M4, which is a particular form ofM3, performs better than the existing two-parameter
Model M1 (score . 80%). In one parameter category, Model M5 performs better than M6
and M2, with score . 50%. As expected from Hawkins et al. (2002) analysis, Model M6
(l ¼ 0.05) performs better than Model M2 (l ¼ 0.2). Here, as expected, the performance
deteriorates with reduction in the number of parameters.
For testing, the validation dataset was utilized to compute RMSE for all the models. A
comparative performance of M3 (three-parameter model) and M5 (one-parameter model) in
calibration and validation is presented in Figures 3 and 4, respectively. Figure 3 shows that,
for Model M3, 90% watersheds have RMSE values less than 8.3mm in calibration, whereas
these are less than 9.3mm in validation, and 50% watersheds have RMSE less than 4.4mm
and 5.2mm in calibration and validation, respectively. Since the validation results are
generally poorer than the calibration results, the closeness of these RMSE values (from
calibration and validation) indicates, in relative terms, a satisfactory model performance. A
similar inference can be drawn from Figure 4. Such an assertion is, however, further checked
on a sample (Q-based) dataset derived differently.
“Q-based” dataset. Figure 5 shows the RMSE values derived using a Q-based dataset in
calibration. The resulting mean RMSE values of Model M1 to M6 are 5.11, 5.30, 5.02, 5.08,
5.16, and 5.17mm, respectively. Thus, the decreasing order of model performance is the
same as for the above “P-based” dataset. As seen from Table 3 showing the results of ranking
and scoring, as above, Model M3 performs the best, and M2 the poorest. Models M4, M1,
M5, and M6 are ranked as 2nd, 3rd, 4th, and 5th, respectively, similar to “P-based” analysis.
Figures 6 and 7 show the RMSE values in calibration and validation, respectively, of
Models M3 and M5. Here, it is noted that these models perform almost equivalently in
Table 3 Percent scores and overall ranking for calibration on both data sets based on RMSE
Data set Calibration M1 M2 M3 M4 M5 M6
“P-based” %score (ranking) 58.13 (iii) 22.82 (vi) 97.62 (i) 82.94 (ii) 51.98 (iv) 36.11 (v)
“Q-based” %score (ranking) 62.1 (iii) 23.61 (vi) 97.82 (i) 79.96 (ii) 46.63 (iv) 39.29 (v)
0
2
4
6
8
10
12
14
0% 20% 40% 60% 80% 100%
Percentage of watersheds less than
R
M
SE
 (
m
m
)
Calibration
Validation
Figure 3 Calibration and validation RMSE comparison of Model M3 for “P-based” dataset
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calibration and validation, in contrast to that in “P-based” analysis. Figure 6 shows nearly
90% and 50% of watersheds to yield RMSE values less than 8.4mm and 4.5mm,
respectively, in calibration of Model M3, while their respective values are 8.3mm and
4.9mm in its validation. Figure 7 also leads to similar results. Thus, the greater closeness
(than that in P-based grouping) of these values reasonably affirms not only a more
satisfactory model performance but also indicates Q-based data grouping to be more
appropriate for model testing than is the P-based data grouping. For relative evaluation of
model performance on a larger dataset, the same analysis was repeated with a full dataset
(only calibration), in what follows.
Performance evaluation on full dataset
Table 4 summarizes the results on a full dataset and Figure 8 compares all the models using
RMSE values. Based on the mean RMSE values of Model M1 to M6, which are 5.10, 5.31,
5.02, 5.09, 5.13, and 5.16mm, respectively, the models can be ranked as follows:
M3 . M4 . M1 . M5 . M6 . M2:
0
2
4
6
8
10
12
14
0% 20% 40% 60% 80% 100%
Percentage of watersheds less than
R
M
SE
 (
m
m
)
Calibration
Validation
Figure 4 Calibration and validation RMSE comparison of Model M5 for “P-based” dataset
0
1
2
3
4
5
6
7
8
9
10
11
12
13
R
M
SE
 (
m
m
)
M1 M2 M3 M4 M5 M6
Figure 5 Statistical comparison of models based on RMSE criteria for “Q-based” dataset of calibration
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It is generally consistent with the above analysis results derived from split data. The
results of model ranking based on their scores (Table 5) indicate Model M3 to perform the
best, and M2 the poorest. Models M4, M1, M5, and M6 rank as second to fifth, respectively.
Figure 9 compares the observed runoff values with the ones computed using Models M1
through M6 for watershed-26018. Model M2 (existing Ia–S relation with l ¼ 0.2) can be
seen to perform most poorly, producing “zero” runoff for most low rainfall-runoff events, in
contrast to other models. Model M1 shows greater scatter than does Model M3 (with the
proposed Ia–S relationship) and M4, indicating the three-parameter model M3 to perform
better than Model M4. In the two-parameter category, Model M4 (a particular form of M3)
performs the best, and it is M5 which performs the best in the one-parameter category. The
last being a one-parameter model, consistent with the existing SCS-CN model, it can be
recommended for field use. Here, it is in order to evaluate these models using the observed
initial abstraction (Ia) data.
Evaluation based on observed Ia data
Hawkins et al. (2002) examined the l variation on a large set of data and recommended
l ¼ 0.05 (Model M6) against the existing l ¼ 0.2 (Model M2). Notably, the Ia–S
relationship was considered the same as Equation (3). Ia values were calculated by summing
0
2
4
6
8
10
12
0% 20% 40% 60% 80% 100%
Percentage of watersheds less than
R
M
SE
 (
m
m
)
Calibration
Validation
Figure 6 Calibration and validation RMSE comparison of Model M3 for “Q-based” dataset
0
2
4
6
8
10
12
14
0% 20% 40% 60% 80% 100%
Percentage of watersheds less than
R
M
SE
 (
m
m
)
Calibration
Validation
Figure 7 Calibration and validation RMSE comparison of Model M5 for “Q-based” dataset
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up the rainfall depth of an event to the point where direct runoff hydrograph began, as shown
in Figure 10 (Hawkins et al. 2002). From these values, S values were computed from
Equation (4). Consistent with the analysis of Hawkins et al. (2002), only those watersheds
containing more than 20 events of P–Ia $ 25.4mm ( ¼ 1 inch) and S $ 0 were selected.
Only 61 out of 84 watersheds satisfied this criterion. In contrast to the “median” l value, the
optimized l value was considered as the representative value for a watershed. Analogous to
the above (Equation (7)), the sum of squares of the difference between observed and
computed values of Ia was minimized in optimization. The results are presented in Table 6.
For Model M1, the resulting standard deviation from the data of 132 ARS watersheds is 0.08
(Hawkins et al. 2002), and it is 0.04 for 61 of these watersheds, approximately half of the
former.
Considering the same RMSE as a goodness-of-fit criterion (Table 7), the models can be
ranked as follows:
M3 . M4 . M5 . M1 . M6 . M2:
Consistent with the results of split and full datasets, Model M2 performed most poorly,
because of a very high l ( ¼ 0.2) value, in conformity with the inference of Hawkins et al.
(2002). Table 8 showing the percentage of overall scores and RMSE-based model ranks
indicate Model M3 to perform the best, and M2 the poorest. Models M4, M5, M1, and M6
rank second to fifth, respectively. Here, it is important to note that M6 proposed by Hawkins
et al. (with l ¼ 0.05) performs poorer than all the other models, except M2.
Table 4 Summary of the optimized parameters on full data set of various models
M1
M2
M3 M4
M5 M6
Statistics l CN CN l a CN l CN CN CN
Min 0.00 13.04 61.56 0.00 21.98 40.12 0.00 54.70 29.23 45.73
Max 1.00 89.97 92.08 50.00 10.00 98.97 2.28 95.95 90.17 88.96
Mean 0.0469.79 81.57 11.06 3.71 82.44 0.79 81.58 73.40 73.74
Median 0.00 73.50 82.22 2.76 2.67 84.44 0.72 82.34 77.41 75.78
STDV 0.13 14.28 6.13 17.02 3.43 13.01 0.60 8.71 12.95 9.37
Skewness 5.62 21.59 21.01 1.62 0.74 21.14 0.61 20.83 21.41 20.95
0
1
2
3
4
5
6
7
8
9
10
11
12
R
M
SE
 (
m
m
)
M1 M2 M3 M4 M5 M6
Figure 8 RMSE comparison of models for full dataset
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Thus, it is evident from the above analysis that the proposed Ia–S relationship (Equation
(6)) performs significantly better than all the other models. It, however, requires the
determination of 3 parameters (CN, l, and a) for application. On the other hand, Model M5,
a one-parameter model, fairly close to Model M3 in performance, is noteworthy and can be
Table 5 Per cent scores and overall rankings for full data set based on RMSE criteria
Models M1 M2 M3 M4 M5 M6
%score 58.13 20.24 100 78.97 54.37 38.1
Overall ranking III VI I II IV V
MODEL 1
0 0.2 0.4 0.6 0.8
Qobs Qobs
0
0.2
0.4
0.6
0.8
Q
co
m
p
Q
co
m
p
0
0.2
0.4
0.6
0.8
Q
co
m
p
Q
co
m
p
0
0.2
0.4
0.6
0.8
Q
co
m
p
0
0.2
0.4
0.6
0.8
Q
co
m
p
0
0.2
0.4
0.6
0.8
MODEL 2
0
0.2
0.4
0.6
0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
Qobs
Qobs Qobs
Qobs
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8
MODEL 3 MODEL 4
MODEL 5 MODEL 6
R = 0.8698 R = 0.8725
R = 0.9030 R = 0. 8726
R = 0.8721 R = 0.8723 
Figure 9 Scatter plot for WS 26018 for models M1, M2, M3, M4, M5, and M6
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Pe = P - Ia
Q = Pe2/(Pe+S)
S = Pe2/Q - Pe
CN = 1000/(10+S)
λ = Ia/S
All Pe > 1 inch
Using median λ value
as the watershed λ
Ia
Pe
Q
P
In
/h
r
Time
Figure 10 Event analysis method (Hawkins et al. 2002)
Table 7 Statistical summary of models based on RMSE criteria for Hawkins et al. (2002) methodology
Models Min Max Mean Median STDV Skewness
M1 3.0148 19.89 9.45 8.21 3.76 0.84
M2 7.0723 11904.08 545.41 99.19 1666.97 5.71
M3 2.5999 13.28 6.63 6.21 2.57 0.60
M4 2.6977 16.76 7.47 7.23 2.99 0.87
M5 3.0164 17.84 7.78 6.72 3.54 0.87
M6 5.0346 2975.24 137.13 23.18 416.18 5.72
Table 6 Summary of the optimized parameters of models based on Hawkins et al. (2002) methodology
M1
M3
M4
Statistics l l a l
Min 0.00 0.09 0.82 0.00
Max 0.21 5.40 7.20 1.08
Mean 0.02 0.63 1.66 0.43
Median 0.01 0.33 1.26 0.35
STDV 0.04 0.83 1.09 0.29
Skewness 3.00 3.90 3.06 0.72
Table 8 Percentage of scores and overall rankings for Hawkins et al. (2002) methodology
Models M1 M2 M3 M4 M5 M6
%score 51.64 17.76 100 74.32 73.77 32.51
Overall ranking IV VI I II III V
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recommended for field applications. In the two-parameter model category, Model M4 (with
varying l and a ¼ 1.5) is somewhat better than M1.
Conclusions
The following conclusions can be derived from the study:
1. The proposed three-parameter Model M3 containing the proposed Ia–S relationship
performed the best of all the other five variants considered in the study, including the
existing Ia–S relationship in Models M1 and M2.
2. Model M3 performed better on “Q-based” datasets than on “P-based” datasets in both
calibration and validation, the best on a full dataset as well as on the data utilizing
observed initial abstraction. On the other hand, the existing SCS-CN method with
l ¼ 0.2 (Model M2) generally performed significantly poorer than all other models.
3. Based on the relative performance on different data sets, Models M5 (with l ¼ 0.3 and
a ¼ 1.5) and M4 (with a ¼ 1.5) in the one- and two-parameter categories,
respectively, can be recommended for field use.
Acknowledgements
The CSIR, Human Resource Development Group, New Delhi, India is gratefully
acknowledged for the assistance provided.
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